\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 129, pp. 1--5.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/129\hfil Decay of solutions]
{Decay of solutions for a plate equation with $p$-Laplacian
and memory term}

\author[W. Liu,  G. Li, L. Hong \hfil EJDE-2012/129\hfilneg]
{Wenjun Liu, Gang Li, Linghui Hong}  % in alphabetical order


\address{Wenjun Liu\newline
College of Mathematics and Statistics, Nanjing University of
Information Science and Technology, Nanjing 210044, China}
\email{wjliu@nuist.edu.cn}

\address{Gang Li\newline
College of Mathematics and Statistics, Nanjing University of
Information Science and Technology, Nanjing 210044, China}
\email{ligang@nuist.edu.cn}

\address{Linghui Hong\newline
College of Mathematics and Statistics, Nanjing University of
Information Science and Technology, Nanjing 210044, China}
\email{hlh3411006@163.com}


\thanks{Submitted April 20, 2012. Published August 15, 2012.}
\subjclass[2000]{35L75, 35B40}
\keywords{Rate of decay; plate equation;  $p$-Laplacian; memory term}

\begin{abstract}
 In this note we show that the assumption on the memory term
 $g$ in Andrade \cite{a1} can be modified to be $g'(t)\leq -\xi(t)g(t)$,
 where $\xi(t)$ satisfies
 $$
 \xi'(t)\leq0,\quad \int_0^{+\infty}\xi(t){\rm d}t=\infty.
 $$
 Then we show that rate of decay for the solution is similar to that 
 of the memory term. 
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

 \section{Introduction}

Consider a bounded domain $\Omega$ in $\mathbb{R}^N$ with smooth boundary
$\Gamma=\partial \Omega$, and study the  solutions to the problem
\begin{gather}
\label{eq1} u_{tt}+\Delta^2u-\Delta_pu+\int_0^tg(t-s)\Delta u(s)
{\rm d}s-\Delta u_t+f(u)=0 \quad \text{in } \Omega\times \mathbb{R}^+ ,\\
\label{eq2} u=\Delta u= 0 \quad \text{on }\Gamma\times \mathbb{R}^+ , \\
\label{eq3} u(\cdot,0) = u_0,\quad  u_t(\cdot,0)=u_1 \quad \text{in } \Omega,
\end{gather}
where $\Delta_pu=\mathrm{div }(|\nabla u|^{p-2}\nabla u )$ is the
$p$-Laplacian operator.

This problem without the memory term models elastoplastic flows.
We refer to \cite{a1} for a motivation and references concerning
the study of problem \eqref{eq1}-\eqref{eq3}. 
We will us the following assumptions:
\begin{itemize}
\item[(A1)] The memory kernel $g$ has typical properties
\begin{equation}
\label{eq4} g(0)>0,\quad l=1-\mu_1\int_0^\infty g(s){\rm d}s>0,
\end{equation}
where $\mu_1>0$ is the embedding constant for $\|\nabla u\|_2^2\leq
\mu_1\|\Delta u\|_2^2$. There exists a constant $k_1>0$ such that
\begin{equation}
\label{eq5} g'(t)\leq-k_1g(t),\quad \forall\ t\geq0.
\end{equation}

\item[(A2)] The forcing term $f$ satisfies
\begin{gather}
\label{eq6} f(0)=0,\quad |f(u)-f(v)|\leq k_2(1+|u|^\rho+|v|^\rho)|u-v|,\quad
 \forall u, v \in \mathbb{R}, \\
\label{eq7} 0\leq \widehat{f}(u)\leq f(u)u,\quad \forall\ u\in \mathbb{R},
\end{gather}
where $k_2$ is a positive constant, $\widehat{f}(z)=\int_0^z f(s){\rm d}s$, 
and 
$$
0<\rho\leq\frac{4}{N-4} \text{ if }  N\geq5 \quad \text{and} \quad 
\rho>0  \text{ if }  1\leq N\leq4.
$$

\item[(A3)] The constant $p$ satisfies
\begin{equation}
\label{eq9} 2\leq p\leq\frac{2N-2}{N-2} \text{ if }  N\geq3 \quad
\text{and} \quad p\geq2  \text{ if } N=1, 2.
\end{equation}
\end{itemize}


\begin{theorem}[{\cite[Theorem 2.1]{a1}}] \label{th1}
Assume that {\rm (A1)--(A3)} hold.
\begin{itemize}
\item[(i)] If the initial data $(u_0,u_1)\in(H^2(\Omega)\cap
H_0^1(\Omega))\times L^2(\Omega)$, then problem
\eqref{eq1}-\eqref{eq3} has a unique weak solution 
$$
u\in C(\mathbb{R}^+; H^2(\Omega)\cap H_0^1(\Omega))\cap
C^1(\mathbb{R}^+;L^2(\Omega)).
$$

\item[(ii)] If the initial data 
$(u_0,u_1)\in H_\Gamma^3(\Omega)\times H_0^1(\Omega)$, where 
$$
H_\Gamma^3(\Omega)=\{u\in H^3(\Omega)|u=\Delta u=0 \text{on } \Gamma \},
$$ 
then problem \eqref{eq1}-\eqref{eq3} has a unique strong solution satisfying
$$
u\in L^\infty(\mathbb{R}^+;H_\Gamma^3(\Omega)),
\quad u_t\in L^\infty(\mathbb{R}^+;H_0^1(\Omega)),\quad
 u_{tt}\in L^2(0,T; H^{-1}(\Omega)).
$$

\item[(iii)] In both cases, the energy  $E(t)$  of problem
 \eqref{eq1}-\eqref{eq3} satisfies the decay rate
$$
E(t)\leq CE(0)e^{-\gamma t},\quad t\geq0,
$$
for some $C, \gamma>0$, where 
\begin{equation}
\label{eq8} E(t)=\frac{1}{2}\|u_t(t)\|_2^2+\frac{1}{2}\|\Delta
u(t)\|_2^2+\frac{1}{p}\|\nabla
u(t)\|_p^p+\int_\Omega\widehat{f}(u(t)){\rm d}x.
\end{equation}
\end{itemize}
\end{theorem}

In this  note, we shall extend the above exponential rate of decay  to
the general case, which is similar to that of $g$. 
We use the following assumption which is weaker than \eqref{eq5}.
\begin{itemize}

\item[(A4)] There exists a positive differentiable function $\xi(t)$ such
that 
$$
g'(t)\leq -\xi(t)g(t),\quad  \forall  t\geq0,
$$ 
and $\xi(t)$ satisfies 
$$
\xi'(t)\leq0,\ \forall\ t>0,\ \int_0^{+\infty}\xi(t){\rm d}t=\infty.
$$
\end{itemize}
Then, we can prove the following main result.

\begin{theorem}\label{th2}
Assume that {\rm (A2)--(A4)} and \eqref{eq4} hold. If the initial data
$(u_0,u_1)\in(H^2(\Omega)\cap H_0^1(\Omega))\times L^2(\Omega)$ or
$(u_0,u_1)\in H_\Gamma^3(\Omega)\times H_0^1(\Omega)$, then the
energy $E(t)$ of problem \eqref{eq1}-\eqref{eq3} satisfies the inequality
\begin{equation}\label{eq10'}
E(t)\leq KE(0)e^{-k\int_{0}^t\xi(s){\rm d}s}, \quad   t\geq 0,
\end{equation}
for some $K, k>0$.
\end{theorem}

\begin{remark} \rm
We note that a similar decay rate was given in  \cite[Theorem 3.5]{m1}.
 However, unlike   \cite[(G2)]{m1} and   \cite[(A1)]{w1}, we do 
not use the condition of $|\frac{\xi'(t)}{\xi(t)}|\le k$ here.
\end{remark}

\begin{remark} \rm
For $\xi(t)\equiv k_1$, \eqref{eq10'} recaptures the exponential decay rate 
in  \cite[Theorem 2.1]{a1}.  For $\xi(t)=a (1+t)^{-1}$, we can get 
polynomial decay rate, which is nt addressed in  \cite{a1}.
\end{remark}

\section{Proof of Theorem \ref{th2}}

Let us first prove the decay property for the strong solution $u$ of
problem \eqref{eq1}-\eqref{eq3}. We modify the perturbed energy method
in  \cite{a1} by using the idea of \cite{l1, m1}.

Assume that condition (A4) holds and define the modified energy, as in \cite{a1},
\begin{align*}
F(t)&=\frac{1}{2}\|u_t(t)\|_2^2+\frac{1}{2}\|\Delta
u(t)\|_2^2+\frac{1}{p}\|\nabla
u(t)\|_p^p+\int_\Omega\widehat{f}(u(t)){\rm d}x\\
&\quad -\frac{1}{2}\Big(\int_{0}^t g(s){\rm d}s\Big)\|\nabla
u(t)\|_2^2+\frac{1}{2}(g\circ \nabla u)(t),
\end{align*}
 where 
$$
(g \circ \nabla u)(t) =\int_{0}^{t}g (t-s)\|\nabla u(t)-\nabla
u(s)\|_{2}^{2}{\rm   d}s.
$$
Then we obtain 
$$
E(t)\leq \frac{1}{l}F(t),
$$ 
and $F(t)$ is decreasing because
\begin{equation} \label{eq10}
\begin{aligned}
F'(t)&=-\|\nabla u_t(t)\|_2^2+\frac{1}{2}(g'\circ \nabla
u)(t)-\frac{1}{2}g(t)\|\nabla u(t)\|_2^2 \\
&\le -\|\nabla u_t(t)\|_2^2-\frac{1}{2}\xi(t)(g\circ \nabla u)(t)\le 0.
\end{aligned}
\end{equation}
Let
$$
\Psi(t)=\int_\Omega u_t(t)u(t){\rm   d}x
$$
and
$$
F_\varepsilon(t)=F(t)+\varepsilon \Psi(t),\quad \forall\ \varepsilon>0.
$$
To obtain the  decay result, we use the following lemmas which are
of crucial importance in the proof.

\begin{lemma}[{\cite[Lemma 4.1]{a1}}]\label{le1} 
There exists $C_1>0$ such that
$$
|F_\varepsilon(t)-F(t)|\leq \varepsilon C_1F(t),\quad \forall t\geq0,\
 \forall\ \varepsilon>0.
$$
\end{lemma}

\begin{lemma}[{\cite[(27) in Lemma 4.2]{a1}}] \label{le2} 
There exist positive constants $C_2, C_3$ such that
\begin{equation} \label{eq11}
 \Psi'(t)\leq-F(t)+C_2\|\nabla u_t(t)\|_2^2+C_3(g\circ \nabla u)(t).
\end{equation}
\end{lemma}

Now, we conclude the proof of the decay property.
Let
$$
\varepsilon_0=\min\big\{\frac{1}{2C_1},\frac{1}{C_2}\big\}.
$$ 
It follows from Lemma \ref{le1} that, for $\varepsilon<\varepsilon_0$,
\begin{equation}\label{eq12} 
\frac{1}{2}F(t)\leq F_\varepsilon(t)\leq\frac{3}{2}F(t),\quad t\geq0.
\end{equation}
By the definition of $F_\varepsilon(t)$, \eqref{eq10} and \eqref{eq11}, 
we obtain
\begin{equation} \label{eq13}
\begin{aligned}
\xi(t)F'_\varepsilon(t)
&=\xi(t)F'(t)+\varepsilon \xi(t)\Psi'(t) \\
&\leq -\xi(t)\|\nabla u_t(t)\|_2^2-\frac{\xi^2(t)}{2}(g\circ \nabla
u)(t)-\varepsilon\xi(t)F(t) \\
&\quad +\varepsilon C_2\xi(t)\|\nabla u_t(t)\|_2^2+\varepsilon
C_3\xi(t)(g\circ \nabla u)(t) \\
&\leq -(1-\varepsilon C_2)\xi(t)\|\nabla
u_t(t)\|_2^2-\varepsilon\xi(t)F(t)+\varepsilon C_3\xi(t)(g\circ \nabla
u)(t) \\
&\leq-\varepsilon\xi(t)F(t)+\varepsilon C_3\xi(t)(g\circ \nabla
u)(t) \\
&\leq-\varepsilon\xi(t)F(t)-2\varepsilon C_3F'(t).
\end{aligned}
\end{equation}
We set
$$
L(t)=\xi(t)F_\varepsilon(t)+2\varepsilon C_3F(t).
$$
Then, $L(t)$ is equivalent to $F(t)$. In fact, we have
$$
L(t)\le \xi(0)F_\varepsilon(t)+2\varepsilon C_3F(t)\le
\Big(\frac{3}{2}\xi(0)+2\varepsilon C_3\Big)F(t)
$$
and
$$
L(t)\ge \frac{1}{2}\xi(t)F(t)+ 2\varepsilon C_3F(t)\ge 2\varepsilon C_3F(t).
$$
Since  $F(t)\ge l E(t)\ge 0$ and $\xi'(t)\le 0$, from \eqref{eq12}
and \eqref{eq13} we obtain
\begin{equation} \label{eq14}
\begin{aligned}
L'(t)&=\xi'(t)F_\varepsilon(t)+\xi(t)F'_\varepsilon(t)+2\varepsilon C_3F'(t) \\
&\leq \xi(t)F'_\varepsilon(t)+2\varepsilon C_3F'(t) \\
&\leq -\varepsilon\xi(t)F(t)\leq-\varepsilon k\xi(t)L(t),
\end{aligned}
\end{equation}
where we have used \eqref{eq13} and $k$ is a positive constant.

A simple integration of \eqref{eq14} leads to
\begin{align}\label{eq15}
L(t)\leq L(0)e^{-k\int_{0}^t\xi(s){\rm d}s},\quad \forall\ t\geq 0.
\end{align}
This proves the decay property for strong solutions in
$H_\Gamma^3(\Omega)$.

The result can be extended to weak solutions
by standard density arguments, as in Cavalcanti et al. \cite{c1, c2}.

\subsection*{Acknowledgements}
This work was partly supported by the Tianyuan Fund of Mathematics
(Grant No. 11026211) and the Natural Science Foundation of the
Jiangsu Higher Education Institutions (Grant No. 09KJB110005).

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\end{document}